Consider the function 𝑓 of 𝑥, 𝑦 equals 𝐾 times 𝑥 cubed over 𝑦. If 𝐾 is a constant in the definition of the function 𝑓 and 𝑓 of 𝑚, 𝑛 equals three, what is the value of 𝑓 of two 𝑚, three 𝑛?
At first, it might not seem like we have enough information to solve this problem. But let’s write down what we know: 𝑓 of 𝑥, 𝑦 equals 𝐾 times 𝑥 cubed over 𝑦 and 𝑓 of 𝑚, 𝑛 equals three. We could also say that 𝑓 of 𝑚, 𝑛 equals 𝐾 times 𝑚 cubed over 𝑛. And since we’re interested in 𝑓 of two 𝑚, three 𝑛, we can say that 𝑓 of two 𝑚, three 𝑛 will equal 𝐾 times two 𝑚 cubed over three 𝑛. We can distribute this cube and say two cubed times 𝑚 cubed. So we can say that 𝑓 of two 𝑚, three 𝑛 equals 𝐾 times two cubed times 𝑚 cubed times three 𝑛.
The 𝐾 stays the same. Two cubed equals eight. 𝑚 cubed equals 𝑚 cubed. And then in our denominator, we have three times 𝑛. Can we regroup 𝐾 times eight times 𝑚 cubed over three times 𝑛 to use the information from 𝑓 of 𝑚, 𝑛? If we take out the eight-thirds, we could say that 𝑓 of two 𝑚, three 𝑛 is equal to eight-thirds times 𝐾𝑚 cubed over 𝑛, which is 𝑓 of 𝑚, 𝑛. And we know that that equals three. That means 𝑓 of two 𝑚, three 𝑛 is equal to eight-thirds times three. The three in the numerator and the three in the denominator cancel out, leaving us with eight.
Based on the given information, we can say that 𝑓 of two 𝑚, three 𝑛 equals eight.